1 MPE and Partial Inversion in Lifted Probabilistic Variable Elimination Rodrigo de Salvo Braz...
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MPE and Partial Inversion inLifted Probabilistic Variable Elimination
Rodrigo de Salvo Braz
University of Illinois at
Urbana-Champaignwith
Eyal Amir and Dan Roth
Page 2
Lifted Probabilistic Inference
We assume probabilistic statements such as8 Person, DiseaseP(sick(Person,Disease) | epidemics(Disease)) = 0.3
Typical approach is grounding. We seek to do inference at first-order level,
like it is done in logic. Faster and more intelligible. Two contributions:
Partial inversion: more general technique than previous work (IJCAI '05)
MPE and Lifted assignments
Page 3
Representing structure
sick(mary,measles)
epidemic(measles) epidemic(flu)
sick(mary,flu)
…
… sick(bob,measles) sick(bob,flu)……
… …
sick(P,D)
epidemic(D)
Poole (2003) named these parfactors,
for “parameterized factors”
Atom
Logical variabl
e
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Parfactor
sick(Person,Disease)
epidemic(Disease)
8 Person, Disease sick(Person,Disease), epidemic(Disease))
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Parfactor
sick(Person,Disease)
epidemic(Disease)
8 Person, Disease sick(Person,Disease), epidemic(Disease)),
Person mary, Disease flu
Person mary, Disease flu
Page 6
Joint Distribution
As in propositional case, proportional to product of all factors But here, “all factors” means all instantiations of all parfactors:
P(...) X (p(X)) X,Y (p(X),q(X,Y))
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Inference task - Marginalization
q(X,Y) X (p(X)) X,Y (p(X),q(X,Y))
Marginal on all random variables in p(X):summation over all assignments to all instances of q(X,Y)
Page 8
The FOVE Algorithm
First-Order Variable Elimination (FOVE): a generalization of Variable Elimination in propositional graphical models.
Eliminates classes of random variables at once.
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FOVE
P(hospital(mary)) = ?
sick(mary,measles)
hospital(mary)
sick(mary, D)
D measles
epidemic(measles) epidemic(D)
D measles
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FOVE
P(hospital(mary)) = ?
sick(mary,measles)
hospital(mary)
sick(mary, D)
D measles
epidemic(D)
D measles
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FOVE
hospital(mary)
sick(mary, D)
D measles
epidemic(D)
D measles
P(hospital(mary)) = ?
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FOVE
P(hospital(mary)) = ?
hospital(mary)
sick(mary, D)
D measles
D measles
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FOVE
P(hospital(mary)) = ?
hospital(mary)
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e(D) D1D2 (e(D1),e(D2))
= e(D) (0,0)#(0,0) in assignment (0,1)#(0,1) in assignment
(1,0)#(1,0) in assignment
(1,1)#(1,1) in assignment
Let i be the number of e(D)’s assigned 1:
= i v1,v2 (v1,v2)#(v1,v2) given i
(number of assignments with |{D : e(D)=1}| = i)
Counting Elimination - A Combinatorial Approach
Page 15
It does not work oneliminating class epidemic from(epidemic(D1, Region), epidemic(D2, Region), donations).
In general, counting elimination does not apply when atoms share logical variables.
Here, Region is shared between atoms.
Counting Elimination - Conditions
Page 16
Partial Inversion
Provides a way of not sharing logical variables
e(D,R) D1D2,R e(D1,R), e(D2,R), d )
R e(D,r) D1D2 e(D1,r), e(D2,r), d )
(R is now bound, so not a variable anymore)
R ’d ) = ’d )|R| = ’’d )
Page 17
Partial Inversion, graphically
epidemic(D2,r1)
epidemic(D1,r1)
D1 D2
donations
epidemic(D2,R)
epidemic(D1,R)
D1 D2 donations
epidemic(D2,r10)
epidemic(D1,r10)
D1 D2…
…
Each instance a counting
elimination problem
Page 18
Another (not so partial) inversion
q(X,Y) X,Y (p(X),q(X,Y)) (expensive)
=X,Y q(X,Y) (p(X),q(X,Y)) (propositional)
= X,Y '(p(X))
= X 'Y(p(X))
= X ''(p(X)) (marginal on p(X))
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Another (not so partial) inversion
…q(x1,y1)
p(x1)
q(xn,yn)
p(xn)…
q(X,Y)
p(X)Each instance a
propositional elimination
problem
Page 20
Partial inversion conditions
friends(X,Y), friends(Y,X))Cannot partially invert on X,Y because friends(bob,mary) appears in more than one instance of parfactor.
friends(mary,bob)
friends(bob,mary)
friends(Y,X)
friends(X,Y)
friends(bob,mary)
…X Y
friends(mary,bob)
…
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Summary of Partial Inversion
More general than previousInversion Elimination.
Generates Counting Elimination or Propositional sub-problems.
Cannot be applied to “entangled parfactors”.
Does not depend on domain size.
Page 22
Second contribution: Lifted MPE
In propositional case,MPE done by factors containing MPE of eliminated variables.
A B
C
D
Page 23
MPE
A B
D
B D MPE
0 0 0.3 C=1
0 1 0.2 C=1
1 0 0.5 C=0
1 1 0.9 C=1
In propositional case,MPE done by factors containing MPE of eliminated variables.
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MPE
A B
B MPE
0 0.5 C=1,D=0
1 1.4 C=1,D=1
In propositional case,MPE done by factors containing MPE of eliminated variables.
Page 25
MPE
A
A MPE(B,C,D)
0 0.9 B=0,C=1,D=0
1 0.7 B=1,C=1,D=1
In propositional case,MPE done by factors containing MPE of eliminated variables.
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MPE
MPE
0.9 A=0,B=1,C=1,D=1
In propositional case,MPE done by factors containing MPE of eliminated variables.
Page 27
MPE
Same idea in First-order case But factors are quantified and so are assignments:
p(X) q(X,Y) MPE
0 0 0.3 r(X,Y) = 1
0 1 0.2 r(X,Y) = 1
1 0 0.5 r(X,Y) = 0
1 1 0.9 r(X,Y) = 1
8 X, Y (p(X), q(X,Y))
Page 28
MPE
After Inversion Elimination of q(X,Y):
p(X) q(X,Y) MPE
0 0 0.3 r(X,Y) = 1
0 1 0.9 r(X,Y) = 1
1 0 0.5 r(X,Y) = 0
1 1 0.3 r(X,Y) = 1
8 X, Y (p(X), q(X,Y))
p(X) ’ MPE
0 0.05 8 Y q(X,Y) = 1, r(X,Y) = 1
1 0.02 8 Y q(X,Y) = 0, r(X,Y) = 1
8 X ’(p(X))
Liftedassignment
s
Page 29
MPE
After Inversion Elimination of p(X):
8 X ’(p(X))
’’ MPE
0.009 8 X 8 Y p(X) = 0, q(X,Y) = 1, r(X,Y) = 0
’’()
p(X) ’ MPE
0 0.05 8 Y q(X,Y) = 1, r(X,Y) = 1
1 0.02 8 Y q(X,Y) = 0, r(X,Y) = 1
Page 30
MPE
After Counting Elimination of e:
e(D1) e(D2) MPE
0 0 0.3 r(D1,D2) = 1
0 1 0.9 r(D1,D2) = 1
1 0 0.5 r(D1,D2) = 0
1 1 0.3 r(D1,D2) = 1
8 D1, D2 (e(D1), e(D2))
’ MPE
0.05 938 D1,D2 e(D1)=0, e(D2) = 0, r(D1,D2) = 1
912 D1,D2 e(D1)=0, e(D2) = 1, r(D1,D2) = 1
915 D1,D2 e(D1)=1, e(D2) = 0, r(D1,D2) = 0
925 D1,D2 e(D1)=1, e(D2) = 1, r(D1,D2) = 1
’()
Page 31
Conclusions
Partial Inversion:More general algorithm, subsumes Inversion elimination
Lifted Most Probable Explanation (MPE) same idea as in propositional VE, but with
Lifted assignments: describe sets of basic assignments universally quantified comes from Partial Inversion existentially quantified comes from Counting elimination
Ultimate goal: to perform lifted probabilistic inference in way similar to
logic inference: without grounding and at a higher level.
Page 32